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We examine the effect of bounding the diameter for well-studied variants of the Colouring problem. A colouring is acyclic, star, or injective if any two colour classes induce a forest, star forest or disjoint union of vertices and edges, respectively. The corresponding decision problems are Acyclic Colouring, Star Colouring and Injective Colouring. The last problem is also known as $L(1,1)$-Labelling and we also consider the framework of $L(a,b)$-Labelling. We prove a number of (almost-)complete complexity classifications. In particular, we show that for graphs of diameter at most $d$, Acyclic $3$-Colouring is polynomial-time solvable if $dleq 2$ but NP-complete if $dgeq 4$, and Star $3$-Colouring is polynomial-time solvable if $dleq 3$ but NP-complete for $dgeq 8$. As far as we are aware, Star $3$-Colouring is the first problem that exhibits a complexity jump for some $dgeq 3$. Our third main result is that $L(1,2)$-Labelling is NP-complete for graphs of diameter $2$; we relate the latter problem to a special case of Hamiltonian Path.
For $kgeq 1$, a $k$-colouring $c$ of $G$ is a mapping from $V(G)$ to ${1,2,ldots,k}$ such that $c(u) eq c(v)$ for any two non-adjacent vertices $u$ and $v$. The $k$-Colouring problem is to decide if a graph $G$ has a $k$-colouring. For a family of gr
We prove several results about the complexity of the role colouring problem. A role colouring of a graph $G$ is an assignment of colours to the vertices of $G$ such that two vertices of the same colour have identical sets of colours in their neighbou
A (proper) colouring is acyclic, star, or injective if any two colour classes induce a forest, star forest or disjoint union of vertices and edges, respectively. Hence, every injective colouring is a star colouring and every star colouring is an acyc
A natural way of increasing our understanding of NP-complete graph problems is to restrict the input to a special graph class. Classes of $H$-free graphs, that is, graphs that do not contain some graph $H$ as an induced subgraph, have proven to be an
The Minimum Path Cover problem on directed acyclic graphs (DAGs) is a classical problem that provides a clear and simple mathematical formulation for several applications in different areas and that has an efficient algorithmic solution. In this pape